Quantum Imaging by Coherent Enhancement

Abstract

Conventional wisdom dictates that to image the position of fluorescent atoms or molecules, one should stimulate as much emission and collect as many photons as possible. That is, in this classical case, it has always been assumed that the coherence time of the system should be made short, and that the statistical scaling ∼1= ffiffi t p defines the resolution limit for imaging time t. However, here we show in contrast that given the same resources, a long coherence time permits a higher resolution image. In this quantum regime, we give a procedure for determining the position of a single two-level system and demonstrate that the standard errors of our position estimates scale at the Heisenberg limit as ∼1=t, a quadratic, and notably optimal, improvement over the classical case.

Methodology of Resonant Equiangular Composite Quantum Gates

“…This exponentially suppresses errors in the transition probability to order n, thus p(θ) ≈ 1 over a wide range of θ. Remarkably, the φ that implement this profile can be found in closed form [48] with optimal sequence lengths L = n. More recently, a second approach has emerged [8], motivated by the following observation: as the flat ansatz p(θ) = 1 − O((θ − π) n ) only increases bandwidth indirectly through the suppression order n, better results can be obtained by directly optimizing for bandwidth, while ensuring that the worst-case error I remained bounded.…”

“…Even in single spin systems, the focus of this work, extraordinary richness can be found in the possible forms ofÛ (θ). For example: NMR spectroscopy, where minute chemical shifts θ are made clearer throughÛ (θ) [2,4,5]; Heisenberg-limited quantum imaging, whereÛ (θ) is made sensitive to sub-wavelength position variations θ [6] without aliasing [7,8]; sub-wavelength spatial addressing where arbitrary quantum gatesÛ (θ) with low crosstalk are applied on spin arrays [8][9][10]; quantum phase estimation for atomic clocks [11] or tomography [12], where extremely small drifts θ are amplified by factor L in the gradient ofÛ (θ); error-compensation, where fractional control errors θ are exponentially suppressed likê U (θ) =Û (0) + O(θ poly(L) ) [13][14][15][16][17][18][19][20]; quantum algorithms such as amplitude amplification [21,22] where a com-putationÛ (θ) proceeds with input θ. The discovery of other applications would be expedited if a useful characterization of all achievableÛ (θ) could be found.…”

“…Given some reasonable system-dependent quantum control [23], its realization as a composite gate must be found. Only with rare exceptions [8,9,16] and great effort are optimal, arbitrary length examples found in closed-form. Thus celebrated techniques including gradient ascent algorithms [5] and pseudospectral methods [24,25] formulate this as a systematic optimization problem that can be solved by brute force, but unfortunately with an exponential worst-case runtime O(e L ) for finding optimally short L approximations.…”

Heisenberg scaling of imaging resolution by coherent enhancement

“…The phases that produce such a narrow-band excitation are described in Refs. [13,21] and are derived from Chebyshev polynomials. In essence, these pulse sequences trade small probabilities of excitation ("ripples") in the stopband for an optimally narrow passband, in analogy to Chebyshev filters [cf.…”

Application of Pontryagin's minimum principle to Grover's quantum search problem

“…In the language of control theory, Grover's algorithm can be regarded as a protocol that approximates the optimal control by the bang-bang control with a minimum number of switchings. Our analysis directly applies to the system effectively involving a single qubit [39,42,43], and may provide insight into the problems of higher dimensions [44,45].…”